Paper Reference(s)
6675/01
Edexcel GCE
Further Pure Mathematics FP2
Advanced Level
Wednesday 21 June 2006 Afternoon
Time: 1 hour 30 minutes
Materials required for examination Items included with question papers
Mathematical Formulae (Green) Nil
Candidates may use any calculator EXCEPT those with the facility for symbolic algebra, differentiation and/or integration. Thus candidates may NOT use calculators such as the Texas Instruments TI 89, TI 92, Casio CFX 9970G, Hewlett Packard HP48G.
Instructions to Candidates
In the boxes on the answer book, write the name of the examining body (Edexcel), your centre number, candidate number, the unit title (Further Pure Mathematics FP2), the paper reference (6675), your surname, initials and signature.
When a calculator is used, the answer should be given to an appropriate degree of accuracy.
Information for Candidates
A booklet ‘Mathematical Formulae and Statistical Tables’ is provided.
Full marks may be obtained for answers to ALL questions.
The marks for individual questions and the parts of questions are shown in round brackets: e.g. (2).
There are 9 questions on this paper. The total mark for this paper is 75.
Advice to Candidates
You must ensure that your answers to parts of questions are clearly labelled.
You must show sufficient working to make your methods clear to the Examiner. Answers
without working may gain no credit.
N26318AThis publication may only be reproduced in accordance with Edexcel Limited copyright policy.
©2006Edexcel Limited.
1.Find the values of x for which
5 cosh x – 2 sinh x = 11,
giving your answers as natural logarithms.
(6)
2.The point S, which lies on the positive x-axis, is a focus of the ellipse with equation + y2 = 1.
Given that S is also the focus of a parabola P, with vertex at the origin, find
(a)a cartesian equation for P,
(4)
(b)an equation for the directrix of P.
(1)
3.The radius of curvature of a curve C, at any point on C, is esincos , where is the angle between the tangent to C at P and the positive axis, and 0 .
Taking s = 0 at = 0, find an intrinsic equation for C.
(4)
4.The curve C has equation y = arctan x2, 0 y.
Find, in surd form, the value of the radius of curvature of C at the point where x = 1.
(6)
5.The curve with equation
y = –x + tanh 4x, x 0,
has a maximum turning point A.
(a)Find, in exact logarithmic form, the x-coordinate of A.
(4)
(b)Show that the y-coordinate of A is {23 – ln(2 + 3)}.
(3)
6. Figure 1
The curve C, shown in Figure 1, has parametric equations
x = t – ln t,
y = 4t, 1 t 4.
(a)Show that the length of C is 3 + ln 4.
(7)
The curve is rotated through 2 radians about the x-axis.
(b)Find the exact area of the curved surface generated.
(4)
7. Figure 2
Figure 2 shows a sketch of part of the curve with equation
y = x2 arsinh x.
The region R, shown shaded in Figure 2, is bounded by the curve, the x-axis and the line x = 3.
Show that the area of R is
9 ln (3 + 10) – (2 + 710).
(10)
8.In = , n 0.
(a)Show that, for n 2,
In = xn sinh x – nxn – 1 cosh x + n(n – 1)In – 2.
(4)
(b)Hence show that
I4 = f(x) sinh x + g(x) cosh x + C,
where f(x) and g(x) are functions of x to be found, and C is an arbitrary constant.
(5)
(c)Find the exact value of , giving your answer in terms of e.
(3)
9.The ellipse E has equation + = 1 and the line L has equation y = mx + c, where m > 0 andc > 0.
(a)Show that, if L and E have any points of intersection, the x-coordinates of these points are the roots of the equation
(b2 + a2m2)x2 + 2a2mcx + a2(c2 – b2) = 0.
(2)
Hence, given that L is a tangent to E,
(b)show that c2 = b2 + a2m2.
(2)
The tangent L meets the negative x-axis at the point A and the positive y-axis at the point B, and O is the origin.
(c)Find, in terms ofm, a and b, the area of triangle OAB.
(4)
(d)Prove that, as m varies, the minimum area of triangle OAB is ab.
(3)
(e)Find, in terms of a, the x-coordinate of the point of contact of L and E when the area of triangle OAB is a minimum.
(3)
TOTAL FOR PAPER: 75 MARKS
END
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